Neccessary conditions for two weight inequalities for singular integral operators

Abstract

We prove necessary conditions on pairs of measures $(\mu,\nu)$ for a singular integral operator $T$ to satisfy weak $(p,p)$ inequalities, $1\leq p<\infty$, provided the kernel of $T$ satisfies a weak non-degeneracy condition first introduced by Stein, and the measure $\mu$ satisfies a weak doubling condition related to the non-degeneracy of the kernel. We also show similar results for pairs of measures $(\mu,\sigma)$ for the operator $T_\sigma f = T(f\,d\sigma)$, which has come to play an important role in the study of weighted norm inequalities. Our major tool is a careful analysis of the strong type inequalities for averaging operators; these results are of interest in their own right. Finally, as an application of our techniques, we show that in general a singular operator does not satisfy the endpoint strong type inequality $T : L^1(\nu) \rightarrow L^1(\mu)$. Our results unify and extend a number of known results.